ODE help

First one looks exact to me (divide out the ). Then use the procedure involving potential functions for exact ODEs to find the solution. second is a linear non-homogeneous equation, so your general solution will be the general solution of the associated homogeneous equation plus any particular solution of the non-homogeneous equation. Since you have constant coefficients, you should use the method of undetermined coefficients.

I got where a is a constant
I use dz=partial z/partial x dx + partial z/ partial y dy method
dunno correct or not

by the way what is the definition of homogenous in layman terms thanks
As for the second Q i got y= C1e^3x +C2e^x

The homogeneous solution is just what you get if the right side, typically written as is zero. It is independent of the right side of the fuction, or the inhomogeneous part. In other words, you will get the same homogeneous solution, no matter what is equal to.

Edit...It might help to know the standard form for a Second Order ODE is

Edit 2...This answer is correct so far, but you have only solved for the homogeneous solution. You need to set up, using the method of undetermined coefficients, a solution to the inhomogeneous part. It will be of the form: because you need a solution linearly independent of the solutions you have already gotten.